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Light cone and Weyl compatibility of conformal and projective structures. (English) Zbl 1460.83007

Summary: In the literature different concepts of compatibility between a projective structure \(\mathscr{P}\) and a conformal structure \(\mathscr{C}\) on a differentiable manifold are used. In particular compatibility in the sense of Weyl geometry is slightly more general than compatibility in the Riemannian sense. In an often cited paper [J. Ehlers et al., Gen. Relativ. Gravitation 44, No. 6, 1587–1609 (2012; Zbl 1245.83004)] Ehlers/Pirani/Schild introduce still another criterion which is natural from the physical point of view: every light like geodesics of \(\mathscr{C}\) is a geodesics of \(\mathscr{P}\). Their claim that this type of compatibility is sufficient for introducing a Weylian metric has recently been questioned [A. Trautman, ibid. 4, No. 6, 1581–1586 (2012; Zbl 1243.83019); the first author with A. Trautman, Commun. Math. Phys. 329, No. 3, 821–825 (2014; Zbl 1298.53021)], as reported by E. Scholz [Gen. Relativ. Gravitation 52, No. 5, Paper No. 46, 39 p. (2020; Zbl 1443.83052)]. Here it is proved that the conjecture of EPS is correct.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C10 Equations of motion in general relativity and gravitational theory
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
53Z05 Applications of differential geometry to physics
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